# Simplified Design of Masonry Ring-Beams Reinforced by Flax Fibers for Existing Buildings Retrofitting

^{*}

## Abstract

**:**

## 1. Introduction

#### Masonry Ring-Beams Reinforced by Natural Fibers

## 2. Materials and Methods

#### 2.1. Materials

_{m}is equal to 2.7 [48]. For fiber-reinforced systems in which only materials are certified (type A applications), and for which the most likely failure is considered to be due to composite detachment, the partial factor γ

_{f}is equal to 1.5 [49]. All of the above values are used in both the simplified and the finite element modeling.

#### 2.2. Methods

_{f}is the total number of layers of reinforcing fiber, h

_{b}is the thickness of bricks, h

_{f}is the thickness of each fiber layer, n

_{b}is the number of clay bricks fully or partially compressed, n is the ratio of the modulus E

_{b}of clay bricks to the modulus E

_{f}of the reinforcing flax fiber, d

_{i}is the distance of the center of each fiber layer from the upper edge of the section (numbered from the top of the section).

_{c}is the compressive strength of bricks at the section upper edge and f

_{t}is the tensile strength of the lowest fiber, in the stressed direction.

_{c}and f

_{t}is equal to the design strength of the corresponding material, while the other one is lower.

^{®}, using the well-known Drucker–Prager failure criterion [57], as a smooth version of the Mohr–Coulomb yield surface, suitable for materials that exhibit volumetric plastic deformations. The cohesion c and the friction angle φ are assumed equal to 0.14 MPa and 45°, respectively [58,59,60]. The flax fibers are placed inside the mortar beds as a fiber-reinforced cementitious matrix composite, and are modeled through “tension only” elements (Table 2). No slip is considered possible between bricks and fiber-reinforced cementitious matrix composites, since the bond is sufficient to keep them acting together under the different load stages, as previously explained for the simplified procedure. The FEM model is composed of minimum 1260 and maximum 3180 elements, due to the variation in the number of clay bricks layers. Figure 3 and Figure 4 show the model of beams made of five layers of solid bricks 60 mm thick (therefore with four layers of fiber-reinforced cementitious matrix composite).

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Reinforced masonry beams differently loaded with respect to the bed joints: (

**a**) Loads orthogonal to the mortar layers; (

**b**) Loads parallel to the mortar layers.

**Figure 2.**Cross-section of reinforced masonry ring-beam: geometric parameters. (

**a**) Bending due to loads orthogonal to the bed joints; (

**b**) Bending due to loads parallel to the bed joints.

**Figure 3.**Finite Element Model (FEM): constraints of a beam made by five layers of solid clay bricks 60 mm thick for loading orthogonal to the bed joints.

**Figure 4.**Finite Element Model: constraints of a beam made by five layers of solid clay bricks 60 mm thick for loading parallel to the bed joints.

**Figure 5.**FEM: load-displacement curve for masonry beam reinforced by flax fiber under vertical loads.

**Figure 6.**Distribution of normal stresses [MPa] in the mid-span cross-section of the FEM masonry beam.

**Figure 7.**Resisting bending moments of masonry beams reinforced by flax fibers for loads orthogonal to the bed joints: (

**a**) solid and hollow bricks 60 mm thick; (

**b**) hollow bricks 120 mm thick.

**Figure 8.**FEM: load-displacement curve for a masonry beam reinforced by flax fiber under horizontal loads.

**Figure 9.**Resisting the bending moments of masonry beams reinforced by flax fibers for loads parallel to the bed joints: (

**a**) solid and hollow bricks 60 mm thick; (

**b**) hollow bricks 120 mm thick.

**Figure 10.**Comparison among the resisting moments of ring-beams in several materials: (

**a**) loads orthogonal to the bed joints; (

**b**) loads parallel to the bed joints.

Brick Thickness [mm] | Brick Type | Load Direction and Compressive Strength [MPa] |
---|---|---|

60 | Solid clay brick | In plane: 34.07 |

60 | Solid clay brick | Out of plane: 11.64 |

60 | Hollow clay brick | In plane: 27.38 |

60 | Hollow clay brick | Out of plane: 6.69 |

120 | Hollow clay brick | In plane: 25.62 |

120 | Hollow clay brick | Out of plane: 3.88 |

Type | MOE [MPa] | Tension [MPa] | Density [g/m^{3}] | Equivalent Thickness [mm] | Elongation at Failure [%] |
---|---|---|---|---|---|

Bidirectional | 43,000 | 532 | 430 | 0.267 | 2.34 |

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**MDPI and ACS Style**

Guadagnuolo, M.; Faella, G.
Simplified Design of Masonry Ring-Beams Reinforced by Flax Fibers for Existing Buildings Retrofitting. *Buildings* **2020**, *10*, 12.
https://doi.org/10.3390/buildings10010012

**AMA Style**

Guadagnuolo M, Faella G.
Simplified Design of Masonry Ring-Beams Reinforced by Flax Fibers for Existing Buildings Retrofitting. *Buildings*. 2020; 10(1):12.
https://doi.org/10.3390/buildings10010012

**Chicago/Turabian Style**

Guadagnuolo, Mariateresa, and Giuseppe Faella.
2020. "Simplified Design of Masonry Ring-Beams Reinforced by Flax Fibers for Existing Buildings Retrofitting" *Buildings* 10, no. 1: 12.
https://doi.org/10.3390/buildings10010012